Trying to kill the mathematical proof

SciAm writer John Horgan still defends his 1993 article on the Death of Proof:

“For millennia, mathematicians have measured progress in terms of what they could demonstrate through proofs — that is, a series of logical steps leading from a set of axioms to an irrefutable conclusion. Now the doubts riddling modern human thought have finally infected mathematics. Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false.”

I cited Thurston as a major force driving this trend, noting that when talking about proofs Thurston “sounds less like a disciple of Plato than of Thomas S. Kuhn, the philosopher who argued in his 1962 book, The Structure of Scientific Revolutions, that scientific theories are accepted for social reasons rather than because they are in any objective sense ‘true.’” I continued:

“‘That mathematics reduces in principle to formal proofs is a shaky idea’ peculiar to this century, Thurston asserts. ‘In practice, mathematicians prove theorems in a social context,’ he says. ‘It is a socially conditioned body of knowledge and techniques.’ The logician Kurt Godel demonstrated more than 60 years ago through his incompleteness theorem that ‘it is impossible to codify mathematics,’ Thurston notes. Any set of axioms yields statements that are self-evidently true but cannot be demonstrated with those axioms. Bertrand Russell pointed out even earlier that set theory, which is the basis of much of mathematics, is rife with logical contradictions related to the problem of self-reference… ‘Set theory is based on polite lies, things we agree on even though we know they’re not true,’ Thurston says. ‘In some ways, the foundation of mathematics has an air of unreality.’”

After the article came out, the backlash—in the form of letters charging me with sensationalism – was as intense as anything I’ve encountered in my career.

For your typical naive reader, this just confirms a modern nihilism, as expressed by this comment:

If it is impossible to codify mathematics, and it is possible to state mathematical ideas that are true, but cannot be proven true, then the same applies to every field. Thus, the mere fact that you cannot prove some idea axiomatically proves nothing. People often demand proofs of that sort for moral or philosophical notions, like, say, the existence of God, but the whole exercise is empty. Thanks, Kurt.

Horgan deserved the criticism, and so did Thurston. It is not true that Godel proved that it is impossible to codify mathematics. It would be more accurate to say the opposite.

Russell did find an amusing set theory paradox in 1901, but it was resolved over a century ago. Set theory is not based on anything false.

Thurston was a great genius, and was famous for explaining his ideas informally without necessarily writing rigorous proofs. He was not an expert in the foundations of mathematics.

The quotes are apparently accurate, as he published as essay defending his views, and his record of claiming theorems and failed to publish the details of the proofs.

His essay complains about the theorem that the real numbers can be well-ordered, but there is no constructive definition of such an ordering.

From this and Godel's incompleteness theorem, he concludes that the foundations of mathematics are "shakier" than higher level math. This is ridiculous. I conclude that his understanding of the real number line is deficient. There is nothing shaky about math foundations.

It sounds crazy to question Thurston's understanding of real numbers, because he was a brilliant mathematician who probably understood 3-dimensional manifolds better than anything.

Doing mathematics is like building a skyscraper. Everything must be engineered properly. But the guy welding rivets on the 22nd floor may not understand how the foundations are built. That is someone else's department. So yes, someone can prove theorems about manifolds without understand how the real numbers are constructed.

It is still the case that mathematicians measure progress in terms of what they could demonstrate through a series of logical steps leading from a set of axioms to an irrefutable conclusion. The most famous works of the past 25 years were Fermat's Last Theorem and the Poincare Conjecture. Both took years to be accepted, because of the work needed to follow all those steps.

The idea that "theories are accepted for social reasons" is the modernist disease of paradigm shift theory.

A NY Times Pi Day article says:

Early mathematicians realized pi’s usefulness in calculating areas, which is why they spent so much effort trying to dig its digits out. ...

So what use have all those digits been put to? Statistical tests have suggested that not only are they random, but that any string of them occurs just as often as any other of the same length. This implies that, if you coded this article, or any other, as a numerical string, you could find it somewhere in the decimal expansion of pi.

That is a plausible hypothesis, and you might even find people willing to bet their lives on it. But you will not find it asserted as true in any mainstream math publication, because it has not been proved. Yes, math relies on proof, just as it has for millennia.